# How can I calculate the following limit? $\lim_{n\to\infty}\frac{1}{(\ln n)^{\ln n}}$

Can you give me just a hint to calculate the following limit? $$\lim_{n\to\infty}\frac{1}{(\ln n)^{\ln n}}$$ I am trying using the squeeze rule without success. I tried also l'Hopital but the limit become tougher.

Thank you.

• Take the logarithm. – Martín-Blas Pérez Pinilla Jan 26 '14 at 19:04
• @Charlie Please, instead of writing $->$ by typing ->, write $\to$ by typing \to or \rightarrow. – user93957 Jan 26 '14 at 19:04
• @Aðøbe Ok, I'm sorry. I'll do it from now on. – Charlie Jan 26 '14 at 19:08
• @Charlie No problem! – user93957 Jan 26 '14 at 19:11

$\ln(n) \rightarrow \infty$ as $n \rightarrow \infty$. So, $\ln(n)^{\ln(n)} \rightarrow \infty$ as $n \rightarrow \infty$. Hence the limit is $0$.
• Please, write $\ln$ by typing \ln instead of $ln$ made by typing ln. – user93957 Jan 26 '14 at 19:06
• @Charlie $\infty^\infty$ is not an indeterminate form. – user93957 Jan 26 '14 at 19:14